In chapter 4 we have successfully applied the new FMM
to crack problems for Laplace's equation and elastostatics and
have shown that the new FMM is faster than the original FMM in
these problems. The improvement has been found to be particularly
remarkable in problems including many cracks.

In chapter 4 we have investigated improvements of the M2L
operations. But we did not implement techniques to reduce M2M and
L2L translations. However, the improved implementation of M2M and
L2L translations is not very difficult and will not be as
effective as the M2L.

The fast multipole algorithm employed in this thesis is
somewhat simpler but may not necessarily be as efficient as the
so-called ``adaptive algorithm'' used in Cheng et
al.[11]. Their adaptive algorithm improves the
efficiency by replacing some of direct evaluations used in the
present implementation by either multipole moments or local
expansions. The algorithm used in this thesis may not be
very efficient in problems, where the distribution of boundary
elements is far from uniform. Hence, we plan to implement
``adaptive algorithm'' in the future work.

In the future work we plan to apply the new FMM and Rokhlin's
diagonal form to dynamic problems. The new FMM for Helmholtz's
equation based on the integral representation for the fundamental
solution of Helmholtz's equation has been proposed by Greengard
et al.[34]. The new FMM for Helmholtz's equation can be
extended to elastodynamics straightforwardly. Rokhlin's diagonal
form seems to work better in high frequency problems than the new
FMM, although it has numerical instabilities in low frequency
problems. Therefore we must make the proper use of the new FMM
and Rokhlin's diagonal form according to the magnitude of the
wave number and the scale of a problems. At the final stage of
applications of FM-BIEM to dynamic problems we plan to use the
hybrid FMM obtained by combination of the new FMM and the
diagonal form.